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Introduction to Trees

Different Shapes and Characteristics of Trees

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It is possible to represent arithmetic, parenthesized expressions using a
tree. If a node is an operator, such as a plus or a division sign, then
each of the two children must be either a number or an expression which
will evaluate to a number. In other words, the two children of an operator
will be its operands.

Figure %: Simple Arithmetic Tree

The above represents
(3 + 4)
.

Problem :
Convert the following expression into such a tree:
((3 + 4)*5)/6

The basic procedure is to determine which operations can be done first (that
is, those that are not dependent on any other operations). Make trees for
those, and then continue this process using the newly formed trees as
operands.

Figure %: Solution 1

Problem :
Convert the following expression into such a tree:
3 + 4*(5/6)

Figure %: Solution 2

Problem :
How could you use this tree representation to devise a scheme to represent
the expressions without using any parentheses. Hint: Consider a the
different sorts of traversals. See the recursion
SparkNote
for information on tree traversals.

If you a post-order traversal, for example, you can create an expression
which is unambiguous and does not use parentheses. In math, this form is
called postfix notation. The way that it can be unambiguously resolved is
that whenever you hit an operator, the two operands for it will be
immediately preceding it. For example:

2 3 4 + *

means add the 3 and the 4 and then multiply by 2. Its parenthesized
equivalent is:
2*(3 + 4)